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Pré-Publication, Document De Travail Année : 2015

Emergence of a collective crystal in a classical system with long-range interactions

Résumé

A one-dimensional long-range model of classical rotators with an extended degree of complexity, as compared to paradigmatic long-range systems, is introduced and studied. Working at constant density, in the thermodynamic limit one can prove the equivalence with the Hamiltonian Mean Field model (HMF) and $\alpha$-HMF: a second order phase transition is indeed observed at the critical energy threshold $\varepsilon_c=0.75$. Conversely, when the thermodynamic limit is performed at infinite density (while keeping the length of the hosting interval $L$ constant), the critical energy $\varepsilon_c$ is modulated as a function of $L$. At low energy, a self-organized collective crystal phase is reported to emerge. According to numerics, the crystal phase is split into two distinct regimes depending on the value of the energy per particle $\varepsilon$: for small $\varepsilon$, particles are exactly located on the lattice sites. At variance, above a given energy threshold $\varepsilon{*}$, such correspondence is broken and particles can travel from one site to the other. However, $\varepsilon{*}$ does not signal a genuine transition but rather reflects the finite time of observation, as imposed in the simulations. Transport between adjacent crystal sites occurs at an exponentially small rate, and a sufficiently large time, even extraordinarily long, has to be allowed for the mixing to take place at small energies. The equilibrium one particle density function is computed by maximizing the entropy, and returns an analytical insight into the scrutinized phenomena.
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Dates et versions

hal-01148798 , version 1 (05-05-2015)
hal-01148798 , version 2 (02-09-2015)

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  • HAL Id : hal-01148798 , version 1

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Alessio Turchi, Duccio Fanelli, Xavier Leoncini. Emergence of a collective crystal in a classical system with long-range interactions. 2015. ⟨hal-01148798v1⟩
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